Positive Solutions for a Class of Two-term Fractional q-derivative Boundary Value Problems with an Integral Condition

Abstract We deduce different conditions to ensure positive solutions for a class of nonlinear two-term Riemann-Liouville fractional q -derivative boundary value problems which depends on a parameter and an integral condition. As a first step, we start by bringing this class of problems through a q -integral equation where its kernel is explicitly identified and its most characteristic properties are analysed. We then use an upper and lower solutions method, an appropriate cone and other techniques to obtain different types of conditions that ensure the existence of a positive solution to the class problems under study. Specific Lipschitz conditions are also identified to ensure the uniqueness of a continuous and positive solution. Finally, using a non-negative continuous concave functional on a cone and comparing the images it produces with the norm of potential solutions to the class of problems, other conditions are obtained that guarantee the existence of three (different) positive solutions to the class of problems under analysis.